Moiré Modulation of Van Der Waals Potential in Twisted Hexagonal Boron Nitride

When a twist angle is applied between two layered materials (LMs), the registry of the layers and the associated change in their functional properties are spatially modulated, and a moiré superlattice arises. Several works explored the optical, electric, and electromechanical moiré-dependent properties of such twisted LMs but, to the best of our knowledge, no direct visualization and quantification of van der Waals (vdW) interlayer interactions has been presented, so far. Here, we use tapping mode atomic force microscopy phase-imaging to probe the spatial modulation of the vdW potential in twisted hexagonal boron nitride. We find a moiré superlattice in the phase channel only when noncontact (long-range) forces are probed, revealing the modulation of the vdW potential at the sample surface, following AB and BA stacking domains. The creation of scalable electrostatic domains, modulating the vdW potential at the interface with the environment by means of layer twisting, could be used for local adhesion engineering and surface functionalization by affecting the deposition of molecules or nanoparticles.

L ayered materials (LMs) are promising both for device applications and for the exploration of fundamental physics. 1 In graphene and related materials (GRMs), such as hexagonal boron nitride (hBN) and transition metal dichalcogenides (TMDs), each layer is bonded by covalent in-plane bonds, whereas weaker van der Waals (vdW) forces hold the layers together. 1 The LM properties can be tuned by controlling the twist angle between layers, producing a spatially modulated interlayer registry, known as moireś uperlattice. 2−4 This can lead to superconductivity 5 and Mott-like insulator states 6 in twisted graphene bilayers, longlived interlayer excitonic states in monolayer (1L) MoSe 2 / WSe 2 heterostructures, 7 and resonant tunneling of graphene Dirac Fermions. 8,9 hBN is a wide-bandgap (∼6 eV) 10 insulating LM with a peculiar set of optical, 11−17 mechanical, 18,19 and electrical properties. 20−22 It is commonly used as an encapsulating material in GRMs. 23 It also gained interest in the context of moiréphysics. For example, scattering near-field optical microscopy (s-SNOM) uncovered the variation of the inplane optical phonon frequencies for different stacking in the moirésuperlattice of a twisted hBN (t-hBN). 24 Piezo force microscopy revealed strain gradients along moiréstacking domain boundaries, through piezoelectric coupling to an electric field applied between atomic force microscope (AFM) tip and hBN sample. 19 Electrostatic force microscopy (EFM) and kelvin probe force microscopy (KPFM) were performed on t-hBN (1−20L-BN on top of a thicker >30L flake 20 ), addressing the existence of two opposite permanent out-of-plane polarizations emerging from the moirépattern. 20−22 However, the impact of moirésuperlattices on local vdW interactions in twisted LMs has not been explored so far, to the best of our knowledge.
Here, we investigate the moiréinterlayer modulation of the vdW potential of t-hBN by using tapping mode AFM phaseimaging, a widely used tool for nanoscale force characterization. 25 In tapping mode AFM, the sine of the phase channel is proportional to the energy dissipated in the tip− sample interaction. 26−33 This depends on the tip−sample distance in a way that is specific to the probed force, 26 allowing noncontact (or long-range) vdW forces to be distinguished from other local interactions, such as capillary, surface energy hysteresis, and viscoelasticity forces. 26,34 By tuning the phase channel to the local vdW dissipation, we quantify the dissipated energy and visualize the modulated vdW potential at the top layer−air interface, resulting from the t-hBN moirésuperlattices. We provide a physical interpretation of the nanoscale origin of the vdW dissipation contrast based on analysis of the tip−sample interaction, showing that the Debye force between the neutral tip and interlayer permanent electric dipoles is the principal source of the imaging contrast. We explain this Debye interaction for the two main stacking domains involved in the t-hBN structure, i.e., AB and BA.
AFM phase imaging is a simpler and more reliable way to visualize moirépatterns in t-LMs. Unlike electric force microscopy techniques, such as EFM and KPFM, it does not require any specific sample or tip biasing. This simplifies sample preparation and reduces the possibility of damage.
Weak electrostatic potentials at the interface with the environment are at the origin of numerous phenomena in fields ranging from fluid dynamics 35 to tribology, 36 both at the macroscopic and microscopic level. 36 We find a modulation on the vdW potential at the sample surface in t-hBN and quantify the related energy dissipation, after calibration of the AFM parameters. The fact that such potential can be patterned in scalable domains engineered by twisting provides a tool for functionalization of surfaces. Locally engineered adhesion, periodically spaced anchoring sites for molecules and nanoparticles deposition, and electrostatically patterned substrates for controlled cells stimulation are a few applications that could benefit from our findings.

RESULTS AND DISCUSSION
We use a t-hBN sample consisting of a 2 nm (∼5L) top hBN layer and an 8 nm (∼20L) bottom hBN, Figure 1a, on Si + 285 nm SiO 2 , as described in Methods. The twist angle, θ twist , is defined as the angle between the lattice vectors of the top and bottom hBN flakes. 37,38 We control this by first identifying neighboring flakes cleaved from the same bulk hBN, as determined by inspection of the relative orientation of their faceted edges, and then picking one flake up using the other. 37 θ twist may be tailored by rotating the transfer stage between picking up the first and second flakes. The accuracy of θ twist is limited by the resolution and wobble of the transfer stage (±0.01°and ±0.008°), monitored by tracking the relative orientations of the faceted edges of top and bottom hBN using optical microscopy and AFM.
AFM/phase/KPFM measurements are taken at ∼25°C (RH ∼ 40%), using a Multimode 8 (Bruker) AFM microscope, with Scanasyst Fluid (Bruker, k ∼ 0.7 N·m −1 , v ∼ 150 kHz), Scanasyst Air HR (Bruker, k ∼ 0.4 N·m −1 , v ∼ 130 kHz), 240AC-NG (OPUS, k ∼ 2 N·m −1 , v ∼ 70 kHz), and ASYELEC.01-R2 (Asylum Research, k ∼ 2.8 N·m −1 , v ∼ 75 kHz) cantilevers. To avoid damaging the tips, calibration procedures are performed at the end of the experiments. The deflection sensitivity is obtained by recording 10 force− distance curves on mica (without changing the laser spot position onto the cantilever) and calculating the average inverse slope of the contact region. The cantilevers spring constant is then obtained using the standard thermal tune method. 39 All the AFM images are obtained in tapping mode at ∼0.5−1 Hz scan rate. These are all postprocessed using Gwyddion. 40 Phase imaging theory 25 states that phase contrast is inversely related to the cantilever spring constant. This points to the need of a soft cantilever. Thus, phase images are taken with k ∼ 1 N·m −1 . No phase moirécontrast is obtained for k ∼ 30 N·m −1 (Cantilever: PPP-NCHAuD, Nanosensors). KPFM maps are also taken with soft cantilevers with the sample holder connected to ground.
AFM phase values tend to follow different conventions depending on the AFM microscope brand. Ref 41 summarized all of them. Bruker's microscopes usually set the free phase (i.e., the phase delay between tip oscillation and cantilever excitation when the cantilever is far from the sample 25 ) to 0°, forcing the attractive regime (AR) to correspond to negative phase values and the repulsive regime (RR) to positive ones. Instead, Asylum Research AFM microscopes set the free phase to 90°, with AR (RR) phase values higher (lower) than this. AFM tapping mode force spectroscopy can be performed to verify these definitions. All phase values in the rest of this paper are renormalized from Bruker to Asylum Research convention.
We simultaneously record topography and phase channels in tapping mode AFM. The phase signal can be described in terms of a forced and damped harmonic oscillator model 42 applied to the dynamics of the AFM cantilever. Figure 1b shows the main parameters. If the cantilever driving excitation is represented by a harmonic signal, i.e., A drive sin(ωt) (with A drive the drive oscillation amplitude, ω = 2πν, with v the cantilever first resonance frequency, and t the time), the tip oscillation corresponds to a delayed sinusoidal motion, i.e., A sin(ωt − φ) (with A the tip oscillation amplitude, kept constant to a set-point by the feedback electronics, and φ the phase signal). A drive is related to the free oscillation amplitude A 0 (i.e., the tip oscillation amplitude when the tip is hundreds nm from the sample) as A A Q drive 0 = , with Q the quality factor of the cantilever resonance. 42 Figure 1c is a representative tapping mode AFM topography image of the air/hBN interface. The morphology is flat, with 1.8 nm modulation over a 1 μm × 1 μm scanned area. The corresponding phase image in Figure 1d, instead, has a periodic pattern characterized by triangular domains with a typical dimension ∼200 nm, consistent with previous observations of moirésuperlattices with electrical AFM modes. 20−22 In Methods we provide a direct comparison between our approach and KPFM. Figure 1d is obtained when the tapping mode probe is operated in AR. 25 The AR and RR concepts in AFM phaseimaging can be described in terms of the nonlinear AFM cantilever dynamics. 43 Assuming the tip−sample interaction to be described by a Lennard-Jones force curve, the tip experiences, over the oscillation period T, either attractive (force F < 0) or repulsive (F > 0) interactions, depending on the instantaneous tip−sample distance (see Figure 2a, where oscillation regions characterized by repulsive forces are shaded in light blue).
An average force, F̅ , can be calculated as the integral of the instantaneous force over one tip oscillation period: Approaching the tip to the sample, i.e., decreasing the distance z c of the cantilever chip from the sample (Figure 2b), two probing regimes can be defined: 1) AR, when the tip is far from the sample (z c ≤ A 0 , F̅ < 0); 2) RR, for z c ≪ A 0 , F̅ > 0. Thus, in AR (RR) the AFM cantilever experiences an average negative (positive) deflection, Figure 2b. The AFM phase channel is a useful tool for monitoring/tuning these two probing regimes. AR and RR correspond to phase values φ > 90°and φ < 90°, respectively. 43 It is possible to move from one to the other by modifying A 0 and A. In our case, to visualize the moireś uperlattice via the phase channel, it is necessary to operate in AR. Figure 2c−j plot topography and phase images of the same hBN region in AR and RR. The topography does not provide  any contrast related to the moirésuperlattice in any of the operating regimes. The phase images instead show a pattern of triangular domains only in AR. When the oscillation regime is switched from AR to RR (from φ > 90°to φ < 90°), decreasing A while keeping constant A 0 , the topography is unaltered (Figure 2c,f), while the moirécontrast completely disappears in the phase map (Figure 2d,g). The moirépattern is recovered by restoring the AR imaging parameters ( Figure 2j).
To test the general applicability of our methodology, we image moirésuperlattices in different regions of the same sample, with different cantilevers (spring constant k ∼ 1 N· m −1 ) and scan size, and on a different t-hBN; see Methods.
Further insights can be obtained by introducing the local dissipation energy of the tip−sample interaction. As discussed    interaction. For a sinusoidal oscillation of a cantilever driven at its resonance frequency, the dissipated energy E diss (in one tip oscillation T) and sin φ are linked by: 26 F ts is the total tip−sample interaction and dz/dt the tip speed along the z-axis at time t. Eq 1 can be considered accurate as long as the dissipative phenomenon does not take place in a low-Q(<10) environment. 31 In this case, contributions from   higher cantilever modes should be considered and a sinusoidal oscillation cannot be accepted. 31 Our Q ∼ 150 (derived by fitting the resonance curve) 25 allows us to use eq 1.
Once a phase image is acquired, a dissipation map can be then reconstructed through eq 1, provided a calibration of k, Q, A, and A 0 is done. Figure 2e,h,k are dissipation energy maps obtained from the phase maps of Figure 2d,g,j by applying eq 1. As for the dissipation maps, the triangular domains of the moirésuperlattice are evident only while operating in AR.
Dissipation in AFM measurements can have different origins. 27 In terms of local interactions at the nanoscale, 3 main dissipative mechanisms at the tip−sample junction can be considered: 26 (1) long-range (i.e., no tip−sample mechanical contact) vdW-like forces; (2) short-range surface energy hysteresis; and (3) short-range viscoelasticity. Ultimately, all are characterized by a different force expression when the tip approaches (forward movement) or withdraws (backward movement) to/from the sample surface, resulting in what is usually called a force−distance hysteretic behavior. 48 These differ for their dependence on the minimum tip−sample distance, d min , that can be controlled by adjusting A/A 0. 26,28 Plotting the dissipation energy as a function of A/A 0 enables the identification of the main dissipation channel responsible for the moirésuperlattice contrast of Figure 1d. In practice, the same t-hBN area is scanned several (∼10) times (with no appreciable drift of the image upon consecutive scanning), keeping A 0 constant and decreasing, at each image, A; see Methods for details. Figure  3a plots E diss as a function of A/A 0 . The trend in Figure 3a is typical of long-range vdW forces. 26,28 Since surface energy hysteresis and viscoelasticity emerge from a tip−sample interaction typical of the repulsive regime, we consider them negligible in the operating attractive regime.
Even though the energy dissipation trend excludes shortrange forces as the moiréimaging contrast mechanism, besides vdW forces, capillary forces could, in-principle, contribute to a similar dissipation behavior. 34 In this case, the contribution of capillary forces would result from the presence of an uncontrolled water layer on the sample, due to ambient humidity (all our AFM measurements are in air at RH ∼ 40%).
In order to distinguish between capillary and vdW forces as dissipative mechanisms, we perform AFM force-spectroscopy. In this case, the tip is not oscillated, but approached and withdrawn to/from the sample, while recording the deflection of the cantilever. n = 300 force−distance curves are collected at the center of both "dark" and "bright" domains of a previously acquired phase map, Figure 3b. Hooke's law allows the force on the tip to be quantified by multiplying the measured cantilever deflection by k.
As discussed in Methods, the comparison of phase and KPFM maps on the same region allows the identification of bright (dark) phase domains as AB (BA) stacking domains. We use this domain classification in the following.   While approaching the tip to the sample surface (blue), a zero-force condition can be found for d > d on , for which tip and sample are far enough that any interaction is negligible. Then, as the tip moves closer to the surface, a small step in the force appears (for d = d on ) that can be ascribed to the formation of a capillary bridge between tip and sample. 49−51 At such distances, attractive vdW forces can affect the tip−sample interaction. 48 When the gradient of the total attractive force overcomes the spring constant of the cantilever, a sudden collapse of the tip toward the sample takes place, caused by the so-called snap-in mechanical instability. 48 At this point, the tip contacts the sample entering a repulsive regime, with a force increasing until set-point deflection is reached.
Retracting the tip from the sample (red), in RR, two separated regions can be distinguished, where approach and withdraw curves are not overlapping, signature of hysteresis. In such regions, the dissipation can be calculated as the enclosed area between the approach and withdraw curves. One hysteresis region extends from d on to d off , where d off corresponds to the distance of rupture of the capillary bridge; 49−51 the other from the adhesion point (d adh , F adh ) to the end of the snap-out 48 (i.e., the equivalent of the snap-in, but for the retraction curve). In this region, the nanoscale water bridge is not broken yet (since d < d off ); therefore, any dissipative contribution only results from vdW forces; see Methods for further details.
This allows us to distinguish the contribution of vdW forces from capillary ones. In our measurements, when the tip is oscillating, A 0 is ∼16 nm, Figure 3a. In this case, a maximum tip oscillation ∼32 nm (= 2·A 0 ) is spanned, covering both vdW and capillary interaction regions. These results, for both AB and BA stacking (see Methods), restrict the capillary contribution to ∼20% of the total dissipation. Thus, the origin of the contrast in the moirépatterns in the phase map of Figure 1d is mainly due to a modulation of the interlayer vdW potential in the moirésuperlattice. The extension of the vdW dissipation regime is restricted to the first five-to-ten nm above the top hBN surface, Figure 3d.
VdW forces emerge from the quantum mechanical interaction between permanent or transient electric dipoles between molecules, 52 i.e., the AFM tip apex and the forefront sample atoms. Casimir forces 53 can be ruled out since they are usually detected on a much larger atoms ensemble by using μm radius spheres rather than sharp tips. 54 Thus, 3 vdW interaction classes can be considered: 52 London, Debye, and Keesom. London forces 55 are the consequence of the interaction between two neutral molecules, whose quantum temporary dipole moments come to a close distance (tens of nm). Debye forces 56 affect a neutral molecule interacting with a polar molecule. Keesom forces 57 emerge from the interaction between two polar molecules. All have an attractive energy U vdW ∝ 1/d 6 , where d is the distance between the two parts. 52 Figure 11. (a,c) AFM tapping mode topography and (b,d) corresponding phase images of a t-hBN (0.8 nm/5.7 nm, θ twist = 0.2°). (c,d) are zooms of (a,b). Despite no contrast in the topography maps, a moirésuperlattice is seen in the phase channels. Cantilever: Scanasyst fluid (Bruker, k ∼ 0.7 N·m −1 ). Imaging parameters: A 0 ∼ 9.5 nm, A ∼ 9 nm, free phase ∼86°.
Refs 20−22 suggested that a layer of ferroelectric dipoles is present at the interface between top and bottom hBN, due to the marginal (<1°) θ twist between the two crystal structures. Hence, our moiréphase-image contrast emerges from the Debye dissipative vdW interaction between tip and sample.
The values of dissipation energy related only to the vdW contribution (E diss vdW ) are in Table 1 for AB and BA domains. These show higher average vdW dissipation energy for BA than AB. This can be qualitatively explained in terms of the different Debye interaction between tip and AB or BA domains. While AB and BA stacking domains both have out-  58 Indeed, we observe moirédomains via AFM phase imaging in t-WSe 2 ; see Figure 16. This shows the general applicability of our imaging approach for LMs.

CONCLUSIONS
We observed the spatial modulation of the vdW potential induced by the moirésuperlattice of t-hBN and t-WSe 2 , via tapping mode AFM phase-imaging, without sample or tip biasing. Our tapping mode AFM phase-imaging is a noninvasive probe for the visualization of the interlayer vdW potential in moirésuperlattices, with no external sample perturbations and compatibility with functional electronic devices in air/liquid/vacuum. By tuning the tip−sample force to the attractive regime, where mainly long-range vdW forces are probed, repulsive interactions were discarded, allowing the visualization of two different triangular vdW domains (AB and BA) emerging from the moirésuperlattice. We quantified the vdW interactions on both AB and BA regions, through the proportionality between phase signal and dissipative tip− sample forces, indicating the BA regions as the most dissipative. We discussed the origin of this nanoscale vdW dissipation and related the interaction between tip and interlayer electric dipoles to a Debye vdW force.
The modulation of the electrostatic potential on the samples, the domain extension and their size can be engineered by twisting the layers. This provides a tool in surface functionalization, enabling to locally tune the electrostatic interaction with the environment on a large scale (>1000 μm 2 ), 19,59 while maintaining a nm resolution. Nanopatterning is an important and diverse research topic continuously enriched by different approaches. 60 Of particular relevance is high-spatial resolution combined with large scale patterning (see, e.g., refs 59−61). LM twisting results in nanopatterning of the interlayer bonding, with periodical domains whose size is tuned by the twist angle. 62 Our results indicate that the twist also results into nanopatterning of the electrostatic field at the sample/environment interface. The modulation produces a local field nanopatterning with the periodicity and tunability of the moirépattern. We can then foresee that moirésuperlattices in insulating and semiconducting LMs could complement already known patterning techniques by lifting the requirement for any sample pretreatment, as for chemical-assisted patterning, 60 or the need for external fields, as in field-assisted patterning. 60 exfoliating bulk hBN (B-hBN) crystals, grown at high pressure and temperature in a barium boron nitride solvent, 63 onto Si + 90 nm SiO 2 by micromechanical cleavage (MC). In order to control θ twist , either large flakes (>50 μm) selectively torn during transfer 37 or neighboring hBN flakes cleaved   Table 1). The related energy difference Δ 2 is also reported. (c) Schematic of the main energies considered in (a,b) for dark (BA) and bright (AB) domains. The vdW dissipation contribution is in green and the capillary dissipation in violet. The main energy values are also shown.

Sample Preparation and Raman Characterization. t-hBN samples are prepared by first
from the same bulk crystal during MC 20 are identified by studying the orientation of their faceted edges using optical microscopy. 64 t-hBN samples with controlled interlayer rotation are then fabricated using polycarbonate (PC) stamps. 65 First, a PC film on polydimethylsiloxane (PDMS) is brought into contact with the substrate with hBN flakes at 40°C using a micromanipulator, so that the contact front between stamp and substrate covers part of one flake or one of two adjacent flakes exfoliated from the same crystal on the tape. Stamps are then retracted, and the material in contact with the PC is picked up from the substrate. After picking up the first flake, a controlled θ twist (±0.01°, as determined by the resolution and wobble of the rotation stage) can be applied by rotating the sample stage, before the flake on PC is aligned to the second one and brought into contact at 40°C. The stamp is then retracted and the resulting t-hBN is picked up by PC. t-hBN is then transferred onto Si + 285 nm SiO 2 at 180°C, before the PC residue is removed by immersion in chloroform and then ethanol for 30 min. While Si + 90 nm SiO 2 is used to facilitate the identification of hBN flakes, 66 Si + 285 nm SiO 2 is chosen for further characterization, such as gate dependent electrical measurements.
Ultralow frequency (ULF) Raman spectroscopy may be used in order to estimate the number of layers, N, of hBN by measuring the position of the C mode, 67,68 Pos(C). For N > 5, the shift in Pos(C), ΔPos(C), can be smaller than the spectral resolution (e.g., ΔPos(C) ∼ 0.15 cm −1 between N = 10 and N = 11 vs a resolution of ∼0.6 cm −1 , corresponding to the wavenumber interval between detector pixels for the combination of diffraction grating and CCD used in the measurements). However, as the Raman peaks are represented by multiple data points even for spectrally narrow ULF modes (e.g., >5 data points for the C mode), it is possible to extract their position with accuracy exceeding the spectral resolution of the experimental setup, via spectral fitting. In general, the error of the peak position extracted via fitting is determined by the fitting error, statistical errors arising from spatial variation, CCD noise and errors associated with the registry of pixels relative to the position of peaks.
In order to extract the error of our measurements for Pos(C) due to fitting and statistical variations, a series of ULF Raman spectra are measured on N > 15 L-hBN using a Horiba LabRAM Evolution at 514 nm, with an 1800 l/mm grating and volume Bragg filters with a ∼5 cm −1 cutoff frequency and a 100× objective (NA: 0.9). Figure  4a shows good agreement between fit and experimental data. The error associated with the Lorentzian fitting is ∼0.03 cm −1 , expected to be negligible compared to statistical errors and pixel registry. In order to evaluate the error due to detector noise, lateral variations across the sample surface and other statistical variations, a series of spectra are acquired at different positions on the same hBN flake. A histogram of Pos(C), from 64 different locations is shown in Figure  4b. The mean Pos(C) is ∼52.67 cm −1 , with a standard deviation ∼0.05 cm −1 and a variation range ∼0.25 cm −1 , which compares favorably with the spectral resolution of the system (∼0.6 cm −1 ).
As the spectral resolution of the system used is comparable to the full width half-maximum, FWHM(C) ∼ 1.1 cm −1 , such that the C peak is depicted by <10 pixels, the registry of the CCD pixels is expected to contribute an additional error. To evaluate this, Pos(C) is extracted by fitting spectra acquired from the same position of a N > 15 L-hBN flake, with grating position offset from −3 to +3 cm −1 in 0.5 cm −1 increments, Figure 5.
A range of grating registries are used so that Pos(C) is at the center of two adjacent pixels or between them. The standard deviation of Pos(C), extracted from Lorentzian fitting, is ∼0.06 cm −1 , with a variation range ∼0.27 cm −1 , less than the spectral resolution of the system. The values of the main fitting errors are in Table 2.
As the relative change of Pos(C) reduces with increasing N, 67,68 for the hBN flakes used here the change in Pos(C) ∼ 0.15 cm −1 between N = 10 and 11 is comparable with the total fitting error ∼ ± 0.15, allowing N to be determined ±1 layer for N < 11. Figure 6 shows the same analysis for the hBN E 2g mode ∼1366 cm −1 . 69−71 The errors associated with fitting it to a Lorentzian, from statistical variation, and pixel registry are summarized in Table 3. The differences compared to Table 2 are due to an increase in FWHM and intensity (relative to the background) for the E 2g mode relative to C. Figure 7 plots the Raman spectra of the 2 and 8 nm hBN flakes, of the resulting t-hBN and the starting B-hBN on Si + 285 nm SiO 2 . Pos(C) = 52.5 ± 0.14 cm −1 for the 8 nm flake, t-hBN and B-hBN, with FWHM(C) = 1 ± 0.2 cm −1 , whereas Pos(C) = 50.1 ± 0.14 cm −1 for the 2 nm flake. Pos(C) can be used to determine N, for N > 2 as 67,72,73 with c the speed of light in cm s −1 , μ = 6.9 × 10 −27 kg Å −2 the mass of one layer per unit area and α ⊥ the interlayer coupling. 67,72,73 In B-hBN, C Pos( ) 52.5 ± 0.14 cm −1 . From this we can derive α ⊥ = 1.69 × 10 18 N m −3 . We then use it in eq 2, and get N = 5 ± 1 for the 2 nm thick flake and N > 10 for the 8 nm one. Figure 7b gives Pos(E 2g ) = 1366 ± 0.2 cm −1 with FWHM(E 2g ) = 8.1 ± 0.2 cm −1 for 8 nm, t-hBN, and B-hBN, whereas FWHM(E 2g ) = 9.8 ± 0.2 cm −1 for the 2 nm flake. The peak broadening ∼1.7 cm −1 in the 2 nm flake can be attributed to strain variations within the laser spot, as thinner flakes conform more closely to the roughness of the underlying SiO 2 . This is also confirmed by the higher RMS roughness of the 2 nm flake (∼0.6 nm) as measured by AFM, compared to ∼ 0.2 nm for the 8 nm flake and t-hBN. Phase and KPFM Maps.  This interpretation of the origin of the energy dissipation map contrast is also in agreement with such domain identification. Figure  8d−f sketch the structure of AA, AB, BA stacking domains. These different alignments are labeled as in refs 20, 22, 76, 77. Due to a symmetric charge distribution of the nitrogen (N) 2p z orbitals, AA has a zero net electric dipole (Figure 8d). The AB configuration (Figure 8e), instead, shows the distortion of the 2p z orbital of the N atom due to its higher electronegativity, 22 resulting in a downward oriented electric dipole closer to the N atom itself. Figure 8f reports BA stacking, characterized by an electric dipole pointing upward. Figure 8c shows the dissipation map corresponding to Figure  8a,b. By direct comparison, the BA stacking domain can be addressed as the most dissipative. An interpretation of this can be provided based on the AB electric dipole being deeper in the material than the corresponding BA dipole (as shown in Figure  8e,f). Consequently, the vdW force (inversely related to the tip− sample distance) is larger when the tip is probing a BA domain, thus leading to a higher dissipation (see eq 1).
AFM of t-hBN (0.8 nm/5.7 nm, θ twist = 0.2°). Figure 11 reports tapping mode AFM topography and phase maps of a t-hBN with different top and bottom layers' thickness and θ twist (0.8 nm/ 5.7 nm and 0.2°, respectively) than the one discussed in the main text. While the topography maps (Figure 11a,c) do not show any relevant feature, in the phase images a moirépattern can be seen.
Dissipation Maps vs A/A 0 . The data of Figure 12 allow us to derive the characteristic curve of Figure 3a. We do not observe flips in the contrast. This is in accordance with the interpretation we provide of the effect of AB and BA stacking. The dipoles of AB and BA sites have different distances from the surface, being the hBN interlayer distance ∼3 Å. This gap is constant whatever the scanning parameters are. The strength of the interlayer dipoles is constant and independent of the scanning parameters. According to these observations, no flip of the contrast should be expected.
Eq 1 can be rewritten as: 26,28 i k j j j j j y E diss depends on the maximum and minimum distance (d max and d min ) of the tip from the considered interlayers dipoles. 26 Therefore, a thicker top-hBN will necessarily increase d max and d min , decreasing E diss . The thickness of the top layer can affect the formation of the domains itself. 20 Such effect would complicate the possibility to set a reference for experimentally deriving the trend of the dissipation energy with respect to the increasing distance due to a thicker top layer. The dissipated energy does not only depend on the tip−sample distance, but also on the hysteresis coefficient α. The physical origin of this parameter is not unique, since several phenomena can contribute to increase the adhesion in the withdraw curves. In ref 78, an extensive list of possible processes is reported. Among them: formation and rupture of chemical bonds between tip and sample, atom reorientation and dislocation, local rearrangement and displacement of atoms. Likely, this would increase the uncertainty in measurements performed on different samples. The only quantitative comparison possible is then between different domains (AB and BA) of the same sample, with the same AFM cantilever, in the same environmental conditions.
Force−Distance Curves on Both AB/BA Stacking Domains. See Figure 13.
VdW Hysteresis Description. Long-range dissipative forces act upon the tip in the noncontact attractive regime, and are typically represented by a vdW-like distance-dependent expression 26 In eq 3, the (effective) Hamaker constant, H eff , represents the magnitude of the vdW interaction between an AFM tip with radius R and the sample at a distance d. 26,78 α ≥ 0 distinguishes between forward and backward movements of the tip with respect to the sample during one oscillation. 78 If the two tip−sample regimes are equal in magnitude (α = 0), a conservative interaction arises providing no dissipation. The existence of a magnitude difference (α > 0), instead, yields a dissipation ∝ α. 28 H eff in eq 3 corresponds to an effective parameter, taking into account all 3 main interactions between tip and sample-substrate. Since thicknesses are in the few nm range, H eff can be identified for tip-ambient-top hBN, tipambient-bottom hBN, and tip-ambient-substrate systems. 79,80 VdW and Capillary Dissipation Energies. See Figure 14.
Topography and Phase Maps in Different Areas. In Figure  1c,d the topography is characterized by a flat morphology plus several straight lines. These are overlaid onto the panel (b) phase map in Figure 15c, providing a direct visualization of their correlation with the moirépattern (at least in one direction). These lines could be either a real local deformation (∼1 Å), induced by the underneath moirésuperlattice, or an apparent topography, following from a different vdW interaction. When imaging in tapping mode nm-scale samples, such as nanoparticles, DNA or hBN flakes, the vdW force between tip and underneath substrate can influence an apparent AFM-height. We do not always observe these additional lines. As shown in Figure 15e, while the phase channel has a moirésuperlattice, the corresponding topography does not have any moire-related feature. Similar considerations apply to Figure 11 (different sample), where the topography channel does not show any feature immediately related to the probed moireś uperlattice. AFM Phase Imaging of t-WSe 2 . Figure 16 plots tapping mode AFM topography, KPFM and phase images of 1L-WSe 2 /1L-WSe 2 (2.1 nm/2.9 nm, θ twist ∼ 0°) on Si + 285 nm SiO 2 . While the morphology (Figure 16a) does not provide any moirécontrast, the KPFM image has some triangular domains highlighted by black and white dash lines. The same moiréKPFM domains are obtained by tapping mode AFM phase-imaging scanning the zoomed red square reported in (Figure 16b).
Force−Distance Curves on AB/BA Domains of t-hBN (0.8 nm/5.7 nm, θ twist = 0.2°) Sample. Figure 17 plots 10 selected F− d curves (out of 300) measured on the center of both BA ( Figure  17a) and AB (Figure 17b) domains for the t-hBN (0.8 nm/5.7 nm, θ twist = 0.2°) sample of Figure 11. In both cases, approach and withdraw curves do not overlap, giving rise to a hysteresis. The corresponding dissipated energy can be obtained calculating the area in between them. Notably, we get a higher average dissipation for BA domains (∼170 eV) than for AB regions (∼162 eV). The different force values with respect to Figure 13 are due to the use of cantilevers with different stiffness: 0.75 N·m −1 in Figure 13; 2.12 N· m −1 for Figure 17.